Langmuir 1997, 13, 4785-4787
4785
Bond-Counting Approach for Representing Association Effects in the Interfacial Region of Multicomponent Systems S. J. Suresh*,† and V. M. Naik Hindustan Lever Research Centre, Hindustan Lever Ltd., A subsidiary of Unilever, Chakala, Andheri(E), Mumbai 400 099, India Received April 7, 1997X
Explicit expressions for the excess Helmholtz free energy due to hydrogen bonding in multicomponent systems, containing either a solid-liquid or a vapor-liquid interface, have been derived. The approach followed in the present article is based on the works of Levine and Perram (1968), who proposed that the focus should be on the correct counting of H-bonds in the system. This approach is distinctly simpler than the reaction equilibrium theory (RET) or the thermodynamic perturbation theory (TPT), proposed earlier by Suresh and Naik (1996), which focuses on the distribution of chain length of H-bonded oligomers/ clusters. In this letter, we demonstrate that, despite the striking difference in the conceptual physical picture of fluids as visualized by RET/TPT and the bond-counting approach, the expressions for the excess free energy of the system in the interfacial region due to H-bonding, derived using either approaches, turn out to be entirely identical. While the mathematical and physical origins of this coincidence need to be investigated, this finding opens up an opportunity to rigorously and quantitatively incorporate association effects into rigorous, but mathematically complex, multilayer theories for interfacial phenomena in a straightforward and simple manner.
Introduction In a previous paper,4 referred to as Paper-I in the rest of the present letter, we proposed a simple theory for rigorously treating hydrogen-bonding (often termed as ‘chemical’, ‘acid-base’, or ‘associative’) interactions, in addition to dispersion and hard-core repulsive (often termed as ‘physical’) interactions to accurately predict interfacial properties of multicomponent systems containing both associating and nonassociating components. The associating components were assumed to exist in shortlived oligomeric forms, and their chain length distribution was determined by assuming that they exist in a state of chemical equilibrium. The various associated entities and nonassociating molecules were then assumed to interact with each other through purely physical forces, as described by any chosen model for intermolecular interactions near the interface. For the purpose of demonstrating the approach, we selected the monolayer model proposed by Prigogine and Marechal2 to describe the physical interactions. Two approaches were developed to satisfy the chemical equilibrium criteria among the associated oligomers: the reaction equilibrium theory (RET) and the thermodynamic perturbation theory (TPT). RET is based on a simple concept that H-bonding can be described by a series of chemical reactions in which a monomer reacts with an n-mer to form an (n+1)-mer. Unfortunately, the applicability of RET is limited to systems containing at most one associating component whose molecules possess at most one donor site and one acceptor site each. On the other hand, TPT is capable of treating a wide variety of complex systems of interest. TPT is, however, based on complex statistical mechanical cluster diagrams, and its derivation is not very transparent. Both approaches, RET and TPT, suffer from an additional drawback that they are not convenient for incorporating association effects into multilayer theories, such as the † X
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self-consistent-field theory proposed by Scheutjens and Fleer,3 for which, as a general case, only numerical solutions exist. For example, performing self-consistentfield calculations for a system containing a large number of components, in the form of H-bonded oligomers, is a computationally prohibitive task. In the above context, we were in search of an approach, other than TPT and RET, which could satisfactorily account for H-bonding interactions in the monolayer and which could also be easily extended to incorporate association effects into more generalized multilayer theories for interfaces. The theory proposed by Levine and Perram,1 which was based on thermodynamic studies of bulk water molecules, was one such alternative approach. This approach is distinctly simpler than TPT/RET as it focuses on the correct counting of the number of H-bonds present in the system and does not concern itself with the determination of the chainlength or molecular-weight distribution of the oligomers/ clusters formed due to association. On the basis of this approach, Veytsman5 developed simple expressions for the excess free energy due to H-bonding in bulk fluids. In the present Letter, we adopt the same approach for describing H-bonding of fluids near the interface. We have found that, despite the striking differences in the conceptual picture as visualized by RET/TPT and the bond-counting approach, the expressions for the excess free energy due to hydrogen bonding in the monolayer, derived using either approaches, turn out to be entirely identical. This finding is demonstrated in the present Letter. While the mathematical and the physical origin of this coincidence needs to be investigated, this finding opens up an opportunity to rigorously and quantitatively incorporate association effects into more realistic but mathematically complex multilayer theories for interfacial phenomena in a straightforward and simple manner. (1) Levine, S.; Perram, J. W. In Hydrogen Bonded Solvent Systems; Covington, A. K., Jones, P., Eds.; Taylor and Francis: London, 1968.
© 1997 American Chemical Society
4786 Langmuir, Vol. 13, No. 18, 1997
Letters
Theory Consider a system that contains a liquid phase and an adjoining solid/vapor phase. Let there be C components, with ni molecules of component i. A molecule of component i may contain ri segments, Sia acceptor sites, and Sid donor sites. The various donor sites of a molecule of component i are assigned characteristic numbers (λid), with the value of λid ranging from 1 to Sid. Similarly, the various acceptor sites of a molecule of component i are assigned characteristic numbers (λia), with the value of λia ranging from 1 to Sia. The system is divided into several lattice layers, with each layer (z) being parallel to the interfacial plane and containing L segments. The layers that comprise molecules belonging to the liquid phase are denoted by z g 2, while the layer that comprises molecules belonging to the adjoining solid/vapor phase is represented by z ) 1. Each segment in any layer z is surrounded by A segments (coordination number), of which Aω(z,z′) segments are present in layer z′ (ω is the weighting factor). In the spirit of Prigogine-Marechal theory,2 the surface of the liquid is considered as a distinct ‘surface’ phase. At equilibrium, H-bonds are assumed to exist between only two association sites belonging to the surface phase (z ) 2), or between one site located in the surface phase (z ) 2) and another site located in the adjacent solid phase (z ) 1). This assumption is consistent with the parallel layer axiom proposed by Prigogine and Marechal2 for adsorption of polymers at the interface. Consider a subsystem in the interfacial region composed of molecules present in layers z ) 1 and z ) 2. We now attempt to evaluate the excess Helmholtz free energy [F(hbond)] of this subsystem due to the presence of H-bonds, the reference state being the same subsystem in an unassociated form. Let m(λid,i,z),(λja,j,z′) represent the number of H-bonds formed between a donor site with characteristic number λid, present on a molecule of type i located in layer z, and an acceptor site with characteristic number λja, present on a molecule of type j located in layer z′. As in this case, the notation followed throughout this Letter is that the first subscript of any H-bond property refers to the identity of the donor site and the second subscript refers to the identity of the acceptor site. In the mean-field approximation, F(hbond) can be written as follows:
the interface, we apply simple combinatorial considerations to obtain
[ [ [ [ 2
Nd(λdi ,i,z)!
Sid
C
∏∏∏
Ω′ )
2
z)1 i)1 λid)1
∏ ∏ ∏ [m
Nd(λdi ,i,z)!
2
z′)1 j)1 λja)1
∏∏∏
2
z′)1 j)1 λja)1
Na(λja,j,z′)!
∏ ∏ ∏ [m z)1 i)1 λdi )1
Sdi
C
2
C
z)1 i)1 λdi )1 z′)1 j)1 λja)1
C
Sdi
2
C
λid)1
λja)1
d i
a j
d i
2
Ω/Ω′ )
Sdi
C
2
C
where f represent the free energy change upon formation of a H-bond between sites denoted by the corresponding paired-subscripts, Ω denotes the number of ways of distributing the set of H-bonds between the various donor and acceptor sites, and k is the Boltzmann factor. The main difficulty in evaluating Ω is to implement the two requirements of H-bond formation: namely, the participating sites should be adjacent to each other and should satisfy a certain angular orientation. Let us first ignore these requirements and determine the number of ways (Ω′) of distributing the set of H-bonds and later correct the resulting expression to include the two constraints. Extending the line of argument that Veytsman5 presented for the case of bulk fluids to fluids near (2) Prigogine, I.; Marechal, J. J. Colloid Sci. 1950, 7, 122. (3) Scheutjens, J. M. H. M.; Fleer, G. J. J. Phys. Chem. 1980, 84, 178. (4) Suresh, S. J.; Naik, V. M. Langmuir 1996, 12, 6151. (5) Veytsman, B. A. J. Phys. Chem. 1990, 94, 8499.
]
(2)
Sja
P(λ ,i,z),(λ ,j,z′)m ∏ ∏ ∏ ∏ ∏ ∏ z)1 i)1 z′)1 j)1 λdi )1
d i
λja)1
a (λd i ,i,z),(λj ,j,z′)
a j
(3)
where P represents the probability that the pair of sites, as specified by the paired-subscripts, satisfies the spatial and angular orientational requirements. The probability that two given sites are proximal to each other is Aω(z,z′)/L since any given site in layer z is proximal to only those present on the Aω(z,z′) segments that are present in layer z′. Let the probability that a pair of sites is favorably oriented be ξ. P is then given by
P(λdi ,i,z),(λja,j,z′) ) ξ(λdi ,i,z),(λja,j,z′)
a j
kT ln(Ω) (1)
[m(λdi ,i,z),(λja,j,z′)!]
where Nd and Na are the total number of donor and acceptor sites of the types specified by the corresponding subscripts and Nd and Na are the number of unbonded donor and acceptor sites of the types specified by the corresponding subscripts. We now correct the above expression to account for spatial and angular orientational requirements of pairs of sites. The ratio Ω/Ω′ is defined as the probability for a configuration of donor-acceptor pairs contributing to Ω′ to contain only those pairs that satisfy spatial and angular orientational requirements of H-bonding and thus to contribute to Ω. If we neglect correlations between the various types of H-bonds formed in the subsystem, this ratio can be written as a product of partial probabilities (P) as follows:
Sja
∑ ∑ ∑ z′)1 ∑∑ ∑ [m(λ ,i,z),(λ ,j,z′)f(λ ,i),(λ ,j)] z)1 i)1 j)1
×
(λdi ,i,z),(λja,j,z′)!]
Sja
[∏ ∏ ∏ ∏ ∏ ∏ 2
F(hbond) ) 2
]
Sdi
C
]
×
(λdi ,i,z),(λja,j,z′)!]
Na(λja,j,z′)!
Sja
C
Sja
C
Aω(z,z′)
(4)
L
Combining eqs 1-4 and applying Stirling’s rule, we obtain the expression for the change in free energy of the subsystem due to the presence of H-bonds: 2
F(hbond) ) kT
Sid
C
∑∑ ∑
z)1 i)1 λdi )1 2
Sja
C
∑∑∑
z′)1 j)1 λja)1 2
C
Sdi
[ ] [ ]
Nd(λdi ,i,z) ln
Na(λja,j,z′) ln
2
Sja
C
Nd(λid,i,z)
+
Nd(λdi ,i,z)
Na(λja,j,z′)
+
Na(λja,j,z′)
∑ ∑ ∑ ∑ ∑ ∑ m(λ ,i,z),(λ ,j,z′) ×
[(
z)1 i)1 λdi )1 z′)1 j)1 λja)1
ln
d i
a j
)]
e[(f(λi ,i),(λj ,j)/kT) + 1]m(λdi ,i,z),(λja,j,z′) d
a
Nd(λdi ,i,z)Na(λja,j,z′)P(λdi ,i,z),(λja,j,z′)
(5)
As usual we only take the maximum term of the grand
Letters
Langmuir, Vol. 13, No. 18, 1997 4787
canonical partition function. Equivalently, at equilibrium, the free energy of the subsystem must be minimum with respect to the set of H-bonds formed in the subsystem: δF(hbond)/δm(λdi ,i,z),(λja,j,z′) ) 0 for any pair of associating sites, at constant volume and temperature. The important assumption we now make is that volume changes upon formation of H-bonds can be neglected. In other words, an associated oligomer would occupy the same volume as the sum of volumes occupied by all the monomers of the oligomer in nonassociated forms. This approximation is in line with that of the Flory-Huggins theory of polymer solutions, wherein a segment of a polymer molecule and a monomeric solvent molecule are both assumed to occupy the same volume corresponding to one lattice site. Applying the minimization condition to eq 5, we obtain:
Ya(λaj ,j,z′) )
[
Na(λja,j,z′) ) Na(λja,j,z′) 2
1+
2
F(hbond) ) LkT Yd(λdi ,i,z)
]
+
(6)
Nd(λid,i,z)Na(λja,j,z′)
Also, the total number of a specific type of site must equal the sum of unbonded and bonded sites of the same type. Hence:
Nd(λid,i,z) )
) Nd(λdi ,i,z) +
ri LΦ(j,z′)
Na(λja,j,z′) )
∑ ∑ ∑ [m(λ ,i,z),(λ ,j,z′)] 2
) Na(λja,j,z′) +
rj
Sja
z′)1 j)1 λja)1
C
d i
a j
(7a)
Sdi
∑ ∑ ∑ [m(λ ,i,z),(λ ,j,z′)]
z)1 i)1 λid)1
d i
a j
(7b)
where Φ(i,z) is the volume fraction of component i in layer z. Combining eqs 6 and 7, we obtain the equilibrium criteria
Yd(λid,i,z) )
[
Nd(λdi ,i,z) ) Nd(λdi ,i,z)
1+
2
C
Sja
∑∑∑
z′)1 j)1 λja)1
]
∆(λdi ,i,z),(λja,j,z′)Ya(λja,j,z′)Φ(j,z′) rj
-1
(8b)
Sdi
C
Φ(i,z)
[ [
λid)1
ri
Sja
Φ(j,z′)
∑∑∑
z′)1 j)1 λja)1
rj
ln(Yd(λdi ,i,z)) +
ln(Ya(λja,j,z′)) +
1 2 2
]
Ya(λja,j,z′)
1
2
(9)
Lm(λdi ,i,z),(λja,j,z′)
C
C
∑∑ ∑ z)1 i)1 2
a
2
ri
z)1 i)1 λdi )1
∆(λid,i,z),(λja,j,z′) ) e-f(λi ,i),(λj ,j)/kT ξ(λid,i,z),(λja,j,z′)Aω(z,z′) )
LΦ(i,z)
∑∑ ∑
]
∆(λdi ,i,z),(λja,j,z′)Yd(λdi ,i,z)Φ(i,z)
Sdi
Substituting eq 8 into eq 5, we obtain the simple expression for the change in free energy of the subsystem, at equilibrium, due to the presence of H-bonds:
2 d
C
-1
(8a)
Equations 8 and 9 are the key expressions of the present Letter. In the case of systems containing one associating component whose molecules possess one donor and one acceptor site, eqs 8 of the present Letter become equivalent to eq 23 of Paper-I, which was derived using the reaction equilibrium theory. For more complex systems, eqs 8 and 9 of the present Letter become identical to eqs 29 and 28, respectively, of Paper-I, which were derived using the thermodynamic perturbation theory. Conclusions The above results suggest that the two apparently different approaches, the previous one focusing on the distribution of chain length of H-bonded oligomers and the present one focusing on only the counting of H-bonds in the system, yield identical expressions for the excess free energy function for the monolayer due to the presence of H-bonds. While it would be worthwhile to investigate the fundamental origin of this equivalence between the two apparently different theories, the finding is very reassuring, as one can now employ the mathematically simpler H-bond counting approach to incorporate association effects into more generalized multilayer theories for interfaces, such as that proposed by Scheutjens and Fleer,3 for which, as a general case, only numerical solutions exist. This will be demonstrated by us in a forthcoming article. Such an endeavor would not be possible using RET/TPT, since performing self-consistentfield calculations for a system containing a large number of components, in the form of H-bonded clusters, would be a computationally prohibitive task. LA970357P
